What if you were given a circle with a quadrilateral inscribed in it? How could you use information about the arcs formed by the quadrilateral and/or the quadrilateral's angle measures to find the measure of the unknown quadrilateral angles? After completing this Concept, you'll be able to apply the Inscribed Quadrilateral Theorem to solve problems like this one.

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Guidance

An
inscribed polygon
is a polygon where every vertex is on a circle. Note, that not every quadrilateral or polygon can be inscribed in a circle. Inscribed quadrilaterals are also called
cyclic quadrilaterals.
For these types of quadrilaterals, they must have one special property. We will investigate it here.

Investigation: Inscribing Quadrilaterals

Place four points on the circle. Connect them to form a quadrilateral. Color the 4 angles of the quadrilateral 4 different colors.

Cut out the quadrilateral. Then cut the quadrilateral into two triangles, by cutting on a diagonal.

Line up
and
so that they are adjacent angles. What do you notice? What does this show?

This investigation shows that the opposite angles in an inscribed quadrilateral are supplementary. By cutting the quadrilateral in half, through the diagonal, we were able to show that the other two angles (that we did not cut through) formed a linear pair when matched up.

Inscribed Quadrilateral Theorem:
A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.

Example A

Find the value of the missing variable.

by the Inscribed Quadrilateral Theorem.
.

by the Inscribed Quadrilateral Theorem.
.

Example B

Find the value of the missing variable.

It is easiest to figure out
first. It is supplementary with
, so
. Second, we can find
.
is an inscribed angle that intercepts the arc
. Therefore, by the Inscribed Angle Theorem,
.
is supplementary with
, so
.Find the value of the missing variables.

Vocabulary

A
circle
is the set of all points that are the same distance away from a specific point, called the
center
. A
radius
is the distance from the center to the circle. A
chord
is a line segment whose endpoints are on a circle. A
diameter
is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. A
central angle
is an angle formed by two radii and whose vertex is at the center of the circle. An
inscribed angle
is an angle with its vertex on the circle and whose sides are chords. The
intercepted arc
is the arc that is inside the inscribed angle and whose endpoints are on the angle. An
inscribed polygon
is a polygon where every vertex is on the circle.